CN114771877B - Optimal interception guidance method considering navigation error - Google Patents

Abstract

An optimal intercept guidance method considering navigation error, the invention relates to an optimal intercept guidance method. The purpose of the present invention is to solve the existence of navigation errors in existing actual missions. If the interception is carried out according to the nominal design orbit, the terminal interception accuracy may not meet the initial condition requirements of the terminal guidance, resulting in the interception mission ending in failure. question. The process is as follows: 1. Use a one-dimensional search algorithm to find the energy-optimized terminal interception time within the interception time range given by the task, and calculate the pulse applied at the initial time; 2. When the navigation error exists at the initial time, solve The optimal interception moment and the error of the applied pulse at the initial moment, and the corresponding terminal interception error is given at the same time; 3. Determine the corrected pulse amplitude and the analytical relationship between the corrected terminal interception error and the corrected time; 4. Determine the corrected pulse application time or corresponding time frame. The invention is used in the technical field of aircraft guidance and control.

Description

An Optimal Intercept Guidance Method Considering Navigation Errors

technical field

The invention belongs to the technical field of aircraft guidance and control, and in particular relates to an optimal interception and guidance method considering navigation errors.

Background technique

Since the world's first artificial satellite was successfully launched in 1957, mankind has truly entered the space age. In the following decades, with the continuous development of aerospace technology, more and more satellites were sent into space, and played an important role in the fields of communication, navigation, meteorological observation and scientific research, bringing great benefits to human life. Great convenience. At the same time, the application of various satellites in the military field is also developing rapidly. In modern warfare, satellites have become the main source of information on the battlefield because of their high orbital altitude, which are not restricted by national borders and geographical conditions, and are not vulnerable to attacks. In order to establish an advantage in the confrontation, it is necessary to study how to effectively intercept these satellites.

For the optimal interception problem of monopulse energy with a fixed interception starting point and an unfixed interception end point, the traditional method is often to design the nominal orbits of the target and interceptor, and then use a one-dimensional search method to obtain the energy within the mission-constrained time range optimal interception time. However, due to the existence of navigation errors in actual missions, if the interception is carried out according to the nominal design orbit, the terminal interception accuracy may not meet the initial condition requirements of the terminal guidance, resulting in the interception mission ending in failure. Therefore, in the process of interception, it is necessary to reduce the influence of navigation errors by applying corresponding correction pulses.

Contents of the invention

The purpose of the present invention is to solve the existence of navigation errors in existing actual missions. If the interception is carried out according to the nominal design orbit, the terminal interception accuracy may not meet the initial condition requirements of the terminal guidance, resulting in the interception mission ending in failure. problem, and an optimal intercept guidance method considering navigation error is proposed.

The specific process of an optimal intercept guidance method considering navigation error is as follows:

并计算得到初始时刻施加的脉冲

Step 1. After the initial time t ₀ and the nominal orbital parameters of the interceptor and the target vehicle are given, set t ₀ as the starting time of the interception task, and use a one-dimensional search algorithm within the interception time range given by the task Finding the optimal energy terminal interception time

And calculate the pulse applied at the initial moment

作为参考，在初始时刻导航误差存在的情况下，基于最优拦截条件求解最优拦截时刻和初始时刻施加脉冲的误差，同时给出相应的终端拦截误差；Step 2. Using the energy optimal solution obtained in step 1

As a reference, in the case of the navigation error at the initial moment, the error of the optimal interception moment and the applied pulse at the initial moment is solved based on the optimal interception condition, and the corresponding terminal interception error is given at the same time;

Step 3. After the correction pulse application time _t1 is given, the amplitude of the applied correction pulse and the corrected terminal interception error are estimated by analytical method, so as to determine the corrected pulse amplitude, the corrected terminal interception error and the correction time t The analytical relationship of ₁ ;

Step 4: Determine the moment when the correction pulse is applied or the corresponding time range.

Beneficial effects of the present invention:

The present invention designs an optimal interception and guidance method considering navigation error, so as to make up for the deficiency in previous design work only for nominal orbit parameters. This method considers the influence of the navigation error of the target and the interceptor on the energy optimal interception process, and analyzes the estimation method of the correction pulse, and then designs a pulse correction strategy, which is conducive to the smooth realization of the interception task .

The invention proposes an optimal intercept guidance method considering navigation error. In the present invention, based on the optimal interception condition and the relevant partial derivative matrix, the relationship between the optimal interception moment and the error of the pulse applied at the initial moment and the initial navigation error is analytically given. Furthermore, for a given correction pulse application time, an analytical estimation method for correction pulse amplitude and terminal interception error is given. By adopting the method of the invention to solve the energy optimal interception problem under the navigation error, the terminal interception accuracy required by the task can be satisfied only by selecting a suitable correction pulse time. In addition, the present invention is also applicable to the situation where correction pulses need to be applied multiple times, and it is only necessary to use the method of the present invention to calculate the next correction pulse that meets the task requirements after the current correction pulse is executed.

Description of drawings

Figure 1 is a flowchart of the Lambert optimal guidance algorithm considering the navigation error;

Fig. 2 is a schematic diagram of the geometry of the single pulse energy optimal interception problem;

Fig. 3 is a graph showing the relationship between the correction pulse amplitude and the standard deviation of the terminal interception error along with the correction time;

Figure 4 is a diagram of the relationship between the weighted index and the modification time.

Detailed ways

Specific implementation mode one: the specific process of an optimal interception guidance method considering navigation error in this implementation mode is as follows:

The kinetic models used in this method are all two-body models, expressed as:

表示r的一阶导数，

表示v的一阶导数；Among them, μ represents the gravitational constant of the earth, r and v represent the position vector and velocity vector of the J2000 inertial system spacecraft respectively, and |r| represents the size of the corresponding position vector;

Indicates the first derivative of r,

Indicates the first derivative of v;

并计算得到初始时刻施加的脉冲

Step 1. After the initial time t ₀ and the nominal orbital parameters of the interceptor and the target vehicle are given, set t ₀ as the starting time of the interception task, and use a one-dimensional search algorithm within the interception time range given by the task Finding the optimal energy terminal interception time

And calculate the pulse applied at the initial moment

作为参考(拦截时刻

以及相应的脉冲

这两者是对应的，已知

已知便可通过Lambert算法求得

)，考虑在初始时刻导航误差存在的情况下，基于最优拦截条件求解最优拦截时刻和初始时刻施加脉冲的误差(公式13和15)，同时给出相应的终端拦截误差(公式16)；Step 2. Using the energy optimal solution obtained in step 1

For reference (intercept time

and the corresponding pulse

The two are corresponding, known

Known can be obtained by Lambert algorithm

), considering the existence of navigation error at the initial moment, based on the optimal interception condition, the error between the optimal interception moment and the applied pulse at the initial moment is solved (Formulas 13 and 15), and the corresponding terminal interception error is given (Formula 16);

Step 3: After the given correction pulse application time t ₁ , the analytical method is used to estimate the amplitude of the applied correction pulse (evaluated by the standard deviation of the pulse amplitude) and the corrected terminal interception error (measured by the standard deviation of the terminal interception error ), so as to determine the analytical relationship between the corrected pulse amplitude and the corrected terminal interception error and the corrected time _t1 (equations 24 and 25 are the corresponding relationships);

Step 4: Determine the appropriate correction pulse application time or corresponding time range according to the optimal set index or according to the relevant accuracy requirements given by the task.

并计算得到初始时刻施加的脉冲

Specific embodiment 2: The difference between this embodiment and specific embodiment 1 is that after the initial time t ₀ and the nominal orbit parameters of the interceptor and the target at the initial time are given in the step 1, t ₀ is set as the interception task At the starting moment of , use a one-dimensional search algorithm to find the terminal interception moment with optimal energy within the interception time range given by the task

And calculate the pulse applied at the initial moment

The specific process is:

The interceptor can be an intercepting satellite of one's own side running in orbit, or an intercepting missile launched by one's own side into orbit;

The target device is an enemy orbiting satellite or other spacecraft;

Given the initial time t ₀ and the interception time range [t _fmin ,t _fmax ], the nominal orbit parameters of the interceptor and the target at the initial time are given in the J2000 geocentric inertial coordinate system, and the nominal orbit parameters of the interceptor and the target at the initial time Said orbit parameters include the position vector and velocity vector of the interceptor at the initial moment and the position vector and velocity vector of the target at the initial moment;

Among them, the position vector and velocity vector at the initial moment of the interceptor are recorded as r ₁₀ and v ₁₀ respectively, and the position vector and velocity vector at the initial moment of the target are respectively recorded as r ₂₀ and v ₂₀ ;

When any interception moment t _f within the interception time range is given, the target terminal position vector _r2 can be obtained by solving the Kepler equation. Note the initial position vector r ₁ =r ₁₀ , then the initial velocity vector v _t1 required for the transfer orbit can be obtained by solving the Lambert problem, that is, v _t1 =f(r ₁ ,r ₂ ,Δt), where Δt=t _f - t ₀ . That is to say, the amplitude of the pulse applied at the initial moment is a one-variable function of the interception time, that is, Δv ₁ =g(t _f ). Therefore, the minimum pulse applied at the initial moment can be obtained by one-dimensional search interception time t _f . It should be noted that since the initial time is predetermined, it can be assumed that t ₀ =0, then Δt=t _f , so in the present invention, the meanings of the two are considered to be the same, and no distinction is made.

并计算得到初始时刻施加的脉冲

Use a one-dimensional search algorithm to find the optimal energy-optimized terminal interception time within the interception time range given by the task

And calculate the pulse applied at the initial moment

Other steps and parameters are the same as those in Embodiment 1.

并计算得到初始时刻施加的脉冲

具体过程为：Specific embodiment 3: The difference between this embodiment and specific embodiments 1 or 2 is that the one-dimensional search algorithm is used to find the terminal interception time with optimal energy within the given interception time range of the task.

And calculate the pulse applied at the initial moment

The specific process is:

In step 1, the one-dimensional search method used is the segmented golden section method, and the specific solution process is described as follows:

First, divide the interception time range [t _fmin ,t _fmax ] into N subintervals, and use the golden section method to solve the corresponding energy optimal solution in each subinterval

(这两者是对应的，求出

便可通过Lambert算法求得

)；Finally, the energy optimal solution in the entire interception time range can be obtained by finding the minimum value among the obtained N energy optimal solutions

(These two are corresponding, find

can be obtained by Lambert algorithm

);

为第k个子区间内的最优拦截时刻，

为第k个子区间内对应的初始时刻施加的脉冲，

为拦截时间范围[t_fmin,t_fmax]内能量最优的终端拦截时刻，

为拦截时间范围[t_fmin,t_fmax]内能量最优的初始时刻施加的脉冲。In the formula,

is the optimal interception time in the kth subinterval,

is the pulse applied at the corresponding initial moment in the kth subinterval,

is the terminal interception time with optimal energy within the interception time range [t _fmin ,t _fmax ],

is the pulse applied at the initial moment with optimal energy within the interception time range [t _fmin ,t _fmax ].

Other steps and parameters are the same as those in Embodiment 1 or Embodiment 2.

作为参考，考虑在初始时刻导航误差存在的情况下，基于最优拦截条件求解最优拦截时刻和初始时刻施加脉冲的误差(公式13和15)，同时给出相应的终端拦截误差；Embodiment 4: The difference between this embodiment and one of Embodiments 1 to 3 is that the energy optimal solution obtained in Step 1 is used in Step 2

As a reference, considering the existence of navigation error at the initial moment, the error between the optimal interception moment and the applied pulse at the initial moment is solved based on the optimal interception condition (Formulas 13 and 15), and the corresponding terminal interception error is given at the same time;

The specific process is:

In step 2, the optimal intercept condition can be expressed as

in

In the formula, Δv ₁ is the velocity pulse that needs to be applied at the initial moment, Δt is the optimal interception time, v _t1 is the initial velocity vector required for the transfer orbit, and T is the transposition;

At the same time, the target terminal position vector (interception position) r ₂ can be expressed as

r ₂ =F _t r ₂₀ +G _t v ₂₀ (3)

in

和

分别表示目标器初始时刻和拦截时刻的真近点角，μ表示地球的引力常数；F_t和G_t为拉格朗日系数；|r₂|表示目标器终端位置矢量的大小；In the formula, p ₂ represents the semi-radius of the target orbit,

and

respectively represent the true anomaly angle of the target at the initial moment and the interception moment, μ represents the gravitational constant of the earth; F _t and G _t are the Lagrangian coefficients; |r ₂ | represents the size of the target terminal position vector;

Taking the partial derivative of equation (3) with respect to the interception time Δt, we can get

Finally, the partial derivative of the target terminal position vector (interception position) r ₂ with respect to the interception time Δt can be obtained as

in

In the formula, I ₃ represents the third-order identity matrix;

关于拦截时间Δt的偏导数为True anomaly angle at the time of target interception

The partial derivative with respect to the interception time Δt is

In the formula, a ₂ and e ₂ are the semi-major axis and eccentricity of the target orbit, respectively, and E ₂ is the approach angle of the target at the moment of interception;

Suppose the position and velocity navigation error of the interceptor at the initial moment is [δr ₁₀ , δv ₁₀ ], and the position and velocity navigation error of the target at the initial moment is [δr ₂₀ , δv ₂₀ ];

Then in the presence of navigation error, the speed pulse that needs to be applied at the initial moment is

In the formula, r ₁ represents the initial position vector of the interceptor, δ(Δt) represents the error of the optimal interception time, δr ₂ represents the error of the target terminal position vector;

是通过求解Lambert问题的偏导数矩阵得到的(参见Zhang G,Zhou D,Mortari D,et al.Covariance analysis of Lambert's problem viaLagrange's transfer-time formulation[J].Aerospace science and technology,2018,77:765-773.)(已知r₁、r₂、Δt，将r₁、r₂、Δt作为Lambert问题的输入，得到v_t1；进而在已知r₁、r₂、Δt、v_t1的情况下，采用上述论文中的方法可输出

)；

It is obtained by solving the partial derivative matrix of the Lambert problem (see Zhang G, Zhou D, Mortari D, et al. Covariance analysis of Lambert's problem via Lagrange's transfer-time formulation [J]. Aerospace science and technology, 2018, 77: 765- 773.) (R ₁ , r ₂ , Δt are known, and r ₁ , r ₂ , Δt are used as the input of the Lambert problem, and v _t1 is obtained; furthermore, given r ₁ , r ₂ , Δt, v _t1 , Using the method in the above paper, the output

);

δr ₂ can be expressed as

In the formula, Φ ^(t) represents the two-body state transition matrix of the target from the initial moment to the interception moment, and its solution method is relatively mature (a simple method can be found in Reynolds R G. Direct solution of the Keplerian state transition matrix[J ].Journal of Guidance Control and Dynamics,2022.Doi:10.2514/1.G006373.);

At the same time, the two-body state transition matrix can also be expressed in the form of a block matrix

表示目标器拦截时刻位置矢量对初始时刻位置矢量的偏导数，

表示目标器拦截时刻的位置矢量对初始时刻速度矢量的偏导数，

表示目标器拦截时刻的速度矢量对初始时刻位置矢量的偏导数，

表示目标器拦截时刻的速度矢量对初始时刻速度矢量的偏导数；Φ^(t)为6×6的矩阵，

均表示3×3的矩阵；In the formula,

Indicates the partial derivative of the position vector of the target interception moment to the initial moment position vector,

Represents the partial derivative of the position vector at the interception moment of the target to the velocity vector at the initial moment,

Represents the partial derivative of the velocity vector at the interception moment of the target to the position vector at the initial moment,

Represents the partial derivative of the velocity vector at the interception moment of the target to the velocity vector at the initial moment; Φ ^(t) is a 6×6 matrix,

Both represent a 3×3 matrix;

According to the optimal interception condition, we can get

Therefore, the error of the optimal interception time can be obtained as

in

In the formula, K represents the total derivative of the initial velocity vector v _t1 required for the transfer orbit to the optimal interception time Δt;

For convenience of description, remember

In the formula, A ₁ represents the coefficient matrix of δ(Δt) affected by δr ₁₀ , A ₂ represents the coefficient matrix of δ(Δt) affected by δv ₁₀ , A ₃ represents the coefficient matrix of δ(Δt) affected by δr ₂₀ , A ₄ Represents the coefficient matrix of δ(Δt) affected by δv ₂₀ ;

Then the error of the optimal interception time can be rewritten as

δ(Δt)=A ₁ δr ₁₀ +A ₂ δv ₁₀ +A ₃ δr ₂₀ +A ₄ δv ₂₀ (13)

The error of applying the pulse (speed increment) at the initial moment is

Similarly, it can be abbreviated as

δ(Δv ₁ )=B ₁ δr ₁₀ +B ₂ δv ₁₀ +B ₃ δr ₂₀ +B ₄ δv ₂₀ (15)

in

In the formula, B ₁ represents the coefficient matrix of δ(Δv ₁ ) affected by δr ₁₀ , B ₂ represents the coefficient matrix of δ(Δv ₁ ) affected by δv ₁₀ , B ₃ represents the coefficient matrix of δ(Δv ₁ ) affected by δr ₂₀ , B ₄ represents the coefficient matrix of δ(Δv ₁ ) affected by δv ₂₀ ;

In the end, in the presence of navigation errors, the relative position error between the interceptor and the target at the time of interception is

in

表示拦截器在拦截时刻的位置矢量，

表示拦截器在拦截时刻的位置误差矢量，C₁表示δ(Δr_f)受δr₁₀影响的系数矩阵，C₂表示δ(Δr_f)受δv₁₀影响的系数矩阵，C₃表示δ(Δr_f)受δr₂₀影响的系数矩阵，C₄表示δ(Δr_f)受δv₂₀影响的系数矩阵；In the formula

Indicates the position vector of the interceptor at the moment of interception,

Represents the position error vector of the interceptor at the moment of interception, C ₁ represents the coefficient matrix of δ(Δr _f ) affected by δr ₁₀ , C ₂ represents the coefficient matrix of δ(Δr _f ) affected by δv ₁₀ , C ₃ represents the coefficient matrix of δ(Δr _f ) is the coefficient matrix affected by δr ₂₀ , C ₄ represents the coefficient matrix of δ(Δr _f ) affected by δv ₂₀ ;

的求解过程与

的求解过程基本一致，只需要将相应的参数更换为转移轨道的参数即可(公式3-8是基于目标器的参数，即r₂₀，v₂₀和Δt进行推导的，其他参数都可在已知这三个量之后计算得到，而转移轨道对应的这三个参数为r₁₀(或r₁，前面定义了二者是等价的)，v_t1和Δt，将r₂₀，v₂₀和Δt相应替换为r₁₀，v_t1和Δt，再进行公式3-8的推导便可求解得到

)；It should be noted

The solution process and

The solution process is basically the same, you only need to replace the corresponding parameters with the parameters of the transfer orbit (Equation 3-8 is derived based on the parameters of the target, namely r ₂₀ , v ₂₀ and Δt, other parameters can be found in the After knowing these three quantities, the three parameters corresponding to the transfer orbit are r ₁₀ (or r ₁ , the two are equivalent as defined above), v _t1 and Δt, and r ₂₀ , v ₂₀ and Δt Correspondingly replaced by r ₁₀ , v _t1 and Δt, and then the derivation of formula 3-8 can be solved to get

);

Therefore, if the navigation error covariance matrix of the interceptor at the given initial moment is [P _r10 , P _v10 ], and the navigation error covariance matrix of the target is [P _r20 , P _v20 ], then the error variance at the optimal interception time for:

The error standard deviation at the optimal interception moment is:

The error covariance matrix of the pulse applied at the initial moment is

The standard deviation of the applied pulse amplitude at the initial moment is

In the formula, trace() represents the trace of the solution matrix;

The error covariance matrix of the relative position of the terminal is

The standard deviation of the terminal relative distance is

In the formula, E[ ] represents mathematical expectation;

In Step 2 and Step 3, the terminal interception error is specifically the relative distance error between the interceptor and the target at the optimal interception moment.

Other steps and parameters are the same as those in Embodiments 1 to 3.

Specific embodiment five: the difference between this embodiment and one of specific embodiments one to four is that in the step three, after the application time _t1 of the given correction pulse, the analytical method is used to estimate the amplitude of the applied correction pulse (in the form of pulse Amplitude standard deviation evaluation) and the corrected terminal interception error (measured by the terminal interception error standard deviation), so as to determine the analytical relationship between the corrected pulse amplitude and the corrected terminal interception error and the correction time _t1 (Formulas 24 and 25 is the corresponding relationship); the specific process is:

Assume that the position and velocity navigation errors of the interceptor and the target at the time of correction are [δr ₁₁ , δv ₁₁ ] and [δr ₂₁ , δv ₂₁ ], respectively;

At this time, according to the position and velocity navigation errors of the interceptor and the target obtained at the correction time are [δr ₁₁ , δv ₁₁ ] and [δr ₂₁ , δv ₂₁ ] respectively, it can be predicted that the position error of the target at the terminal interception time is

为目标器从修正时刻t₁到拦截时刻t_f的状态转移矩阵；

为目标器拦截时刻位置矢量对修正时刻位置矢量的偏导数矩阵，

为目标器拦截时刻位置矢量对修正时刻速度矢量的偏导数矩阵；In the formula,

is the state transition matrix of the target from the correction time t ₁ to the interception time t _f ;

is the partial derivative matrix of the position vector at the interception time of the target to the position vector at the correction time,

is the partial derivative matrix of the position vector at the interception moment of the target to the velocity vector at the correction moment;

Assuming that the corrected pulse vector applied at time t ₁ is q, the position and velocity navigation errors of the interceptor and the target obtained at the corrected time are [δr ₁₁ , δv ₁₁ ] and [δr ₂₁ , δv ₂₁ ] to calculate the interceptor at The position error at the moment of terminal interception is

为拦截器从修正时刻t₁到拦截时刻t_f的状态转移矩阵；

为拦截器拦截时刻位置矢量对修正时刻位置矢量的偏导数矩阵，

为拦截器拦截时刻位置矢量对修正时刻速度矢量的偏导数矩阵；In the formula,

is the state transition matrix of the interceptor from the modification time t ₁ to the interception time t _f ;

is the partial derivative matrix of the position vector at the interception time of the interceptor to the position vector at the correction time,

is the partial derivative matrix of the position vector at the interception time of the interceptor to the velocity vector at the correction time;

Since the purpose of applying the correction pulse is to make the interceptor and the target at the same position at the moment of interception, that is, the following conditions are met

Then the applied correction pulse can be calculated as

The above formula can be abbreviated as

in

where D ₁ represents the coefficient matrix of q affected by δr ₁₀ ; D ₂ represents the coefficient matrix of q affected by δv ₁₀ ; D ₃ represents the coefficient matrix of q affected by δr ₂₀ ; D ₄ represents the coefficient of q affected by δv ₂₀ matrix; D ₅ represents the coefficient matrix of q affected by δr ₁₁ ; D ₆ represents the coefficient matrix of q affected by δv ₁₁ ; D ₇ represents the coefficient matrix of q affected by δr ₂₁ ; D ₈ represents the coefficient matrix of q affected by δv ₂₁ ;

Then the covariance matrix of the applied correction pulse is

P _q =E[qq ^Τ ]

The standard deviation of the applied correction pulse amplitude is

On this basis, the real initial state of the interceptor, after applying the "inaccurate" initial pulse and correction pulse calculated by the orbit data with navigation error, at the optimal interception time, the real interceptor and target The relative position error of can be expressed as

是拦截器真实的初始状态在施加“不准确”的初始脉冲和修正脉冲后，在拦截时刻的位置误差；In the formula,

is the position error of the interceptor at the moment of interception after the "inaccurate" initial pulse and correction pulse are applied to the real initial state of the interceptor;

Then the covariance matrix of the terminal relative position error after applying the correction is

The standard deviation of the terminal relative distance after applying the correction is

以及计算得到相应修正脉冲幅值标准差σ(q)和修正后的终端拦截误差标准差

In summary, the correction pulse q applied at time t ₁ and the terminal relative position error after correction are obtained

And calculate the standard deviation of the corresponding corrected pulse amplitude σ(q) and the corrected standard deviation of the terminal interception error

对应于施加修正脉冲的情况；In the present invention, the 1σ value of the terminal interception error is used as an index to measure the terminal interception accuracy, wherein σ(Δr _f ) corresponds to the situation where no correction pulse is applied,

Corresponding to the case of applying a correction pulse;

Other steps and parameters are the same as in one of the specific embodiments 1 to 4.

Embodiment 6: The difference between this embodiment and one of Embodiments 1 to 5 is that in the step 4, an appropriate corrective pulse application is determined according to the optimal set index or according to the relevant accuracy requirements given by the task. Moment or the corresponding time range; the specific process is:

和修正脉冲幅值标准差σ(q)的加权指标。根据步骤三中的结果可知

和σ(q)均是修正脉冲施加时刻t₁的一元函数，可表示为In step four, some indicator can be assumed to be terminal intercept error standard deviation

and the weighted index of the standard deviation σ(q) of the corrected pulse amplitude. According to the results in step 3, it can be seen that

and σ(q) are both unary functions of the correction pulse application time t ₁ , which can be expressed as

In the formula, h ₁ () represents the functional relationship between the terminal interception error standard deviation and the correction pulse application time, and h ₂ () represents the functional relationship between the correction pulse amplitude standard deviation and the correction pulse application time;

Therefore, the weighted index J can be designed as

和

是归一化参数，可根据任务需求合理设计使得归一化后两个指标的量级相当；Where J represents the weighted index, w ₁ and w ₂ represent the weight coefficients of the two indexes respectively, and satisfy w ₁ +w ₂ =1,

and

is a normalization parameter, which can be reasonably designed according to the task requirements so that the magnitudes of the two indicators after normalization are equivalent;

Finally, the weighting index can be expressed as a unary function of the moment of applying the correction pulse, that is, J=h ₃ (t ₁ ), and then a one-dimensional search algorithm can be used to determine the moment of applying the correction pulse that makes the weighting index J optimal;

In the formula, h ₃ () represents the functional relationship between the weighting index J and the correction pulse application time t ₁ .

Other steps and parameters are the same as one of the specific embodiments 1 to 5.

Embodiment 7: The difference between this embodiment and one of Embodiments 1 to 5 is that in the step 4, an appropriate corrective pulse application is determined according to a certain index that is set to be optimal or according to the relevant accuracy requirements given by the task. Moment or the corresponding time range; the specific process is:

的精度要求，如设置

的精度上限为

In step four, the relative accuracy requirements for a given task can be considered as

The accuracy requirements, such as setting

has an upper precision limit of

是脉冲修正时刻的一元函数。因此，可通过二分法求解方程

的零根，从而得到满足终端拦截精度要求的修正脉冲施加时刻范围；because

is a unary function of the pulse correction time. Therefore, the equation can be solved by bisection

The zero root of , so as to obtain the corrected pulse application time range that meets the requirements of terminal interception accuracy;

In the formula, h ₁ () represents the functional relationship between the terminal interception error standard deviation and the correction pulse application time.

Other steps and parameters are the same as one of the specific embodiments 1 to 5.

Adopt the following examples to verify the beneficial effects of the present invention:

Embodiment one:

Let the nominal position velocity vector of the interceptor at the initial moment be:

The nominal position and velocity vector of the target at the initial moment is:

The standard deviation of the interceptor's navigation error is

The standard deviation of the navigation error of the target is

初始时刻施加脉冲的标称值为

其脉冲幅值大小为

First, set the initial time t ₀ =0, and the search range of the interception terminal time is [0,14400]s, and then under the nominal state, solve the energy-optimal monopulse interception problem with a fixed interception starting point and an uncertain interception end point. Finally, the nominal value of the optimal interception time is obtained by searching the segmented golden section method

The nominal value of the pulse applied at the initial moment is

Its pulse amplitude is

Considering the existence of the navigation error between the interceptor and the target at the initial moment, the optimal interception time, maneuver pulse amplitude, and terminal interception error standard deviation can be calculated according to the analytical method in step 2, and the Monte-Carlo method is used to shoot the target Simulation was performed 1000 times for verification, and the results are shown in Table 1:

Table 1 Influence of initial navigation error on optimal intercept parameters (1σ)

From the above results, it can be found that, considering the existence of the initial navigation error, if the interception is carried out according to the nominal design result, the standard deviation of the terminal interception error (1σ) is about 50km, and the interception mission is very likely to fail. Therefore, corresponding correction pulses must be applied during the interception process to compensate for the influence of navigation errors on terminal interception accuracy.

First, according to the results in step 3, analyze the influence of different correction pulse application time on the applied pulse amplitude and terminal interception error, the specific situation is shown in Figure 3;

加权系数取为w₁＝w₂＝0.5。则加权指标随修正时刻的变化情况如图4所示；Case 1: Setting normalization parameters

The weighting coefficient is taken as w ₁ =w ₂ =0.5. Then, the change of the weighted index with the correction time is shown in Figure 4;

为了验证所得结果的正确性，采用Monte-Carlo方法打靶仿真1000次，对所得结果进行统计分析，得到修正脉冲幅值标准差为0.0144km/s，修正后的终端拦截误差标准差为13.7425km。可以看出与解析方法所得结果基本一致，证明了本发明中方法的有效性。Figure 4 shows that there is a minimum value point in the weighting index throughout the interception process, so the correction time corresponding to the minimum value of the weighting index obtained by using the golden section method is t ₁ =5333.767s. Using the method in step 3, it can be estimated that the standard deviation of the corrected pulse amplitude applied at this moment is σ(q) = 0.0142 km/s, and the standard deviation of the terminal interception error after correction is

In order to verify the correctness of the obtained results, the Monte-Carlo method was used for 1000 target shooting simulations, and statistical analysis was performed on the obtained results. The standard deviation of the corrected pulse amplitude was obtained to be 0.0144 km/s, and the standard deviation of the corrected terminal intercept error was 13.7425 km. It can be seen that the results obtained by the analytical method are basically consistent, which proves the effectiveness of the method in the present invention.

由于

为拦截时刻t₁的函数，因此，只要求解方程

的零根即可。采用二分法求解得到方程的解为t₁＝6332.827s，即修正脉冲只需在该时刻之后添加，即可满足任务给定的终端拦截精度要求。由本发明中的解析方法可计算得到，在该时刻施加的修正脉冲幅值标准差为σ(q)＝0.0200km/s，修正后的终端拦截误差标准差为

采用Monte-Carlo方法打靶仿真1000次，对所得结果进行统计分析，得到修正脉冲幅值标准差为0.0203km/s，修正后的终端拦截误差标准差为10.0493km。由此说明，本发明所提出的方法是正确有效的。Situation 2: The requirement of setting task for terminal interception accuracy is

because

is a function of the interception time t ₁ , therefore, only the solution of the equation

The zero roots of . The solution of the equation obtained by using the dichotomy method is t ₁ =6332.827s, that is, the correction pulse only needs to be added after this time to meet the terminal interception accuracy requirement given by the task. Can be calculated by the analytical method in the present invention, the correction pulse amplitude standard deviation that applies at this moment is σ (q)=0.0200km/s, the terminal interception error standard deviation after correction is

The Monte-Carlo method was used to simulate 1000 times of target shooting, and the statistical analysis of the obtained results showed that the standard deviation of the corrected pulse amplitude was 0.0203km/s, and the standard deviation of the corrected terminal interception error was 10.0493km. This shows that the method proposed by the present invention is correct and effective.

The present invention can also have other various embodiments, without departing from the spirit and essence of the present invention, those skilled in the art can make various corresponding changes and deformations according to the present invention, but these corresponding changes and deformations are all Should belong to the scope of protection of the appended claims of the present invention.

Claims (1)

1. a kind of optimal intercept guidance method that considers navigation error, it is characterized in that: described method specific process is: Step 1. After the initial time t ₀ and the nominal orbital parameters of the interceptor and the target vehicle are given, set t ₀ as the starting time of the interception task, and use a one-dimensional search algorithm within the interception time range given by the task Finding the optimal energy terminal interception time

And calculate the pulse applied at the initial moment

Step 2. Using the energy optimal solution obtained in step 1

As a reference, in the case of the navigation error at the initial moment, the error at the optimal interception moment and the error of the pulse applied at the initial moment are solved based on the optimal interception condition, and the corresponding terminal interception error is given at the same time; Step 3. After the correction pulse application time _t1 is given, the amplitude of the applied correction pulse and the corrected terminal interception error are estimated by analytical method, so as to determine the corrected pulse amplitude, the corrected terminal interception error and the correction time t The analytical relationship of ₁ ; Step 4. Determine the moment when the correction pulse is applied or the corresponding time range; After the initial moment _t0 and the nominal orbital parameters of the interceptor and the target vehicle are given in the step 1, _t0 is set as the initial moment of the interception task, and the one-dimensional The search algorithm finds the optimal energy terminal interception moment

And calculate the pulse applied at the initial moment

The specific process is: The interceptor can be an intercepting satellite of one's own side running in orbit, or an intercepting missile launched by one's own side into orbit; The target device is an enemy orbiting satellite or other spacecraft; Given the initial time t ₀ and the interception time range [t _fmin ,t _fmax ], the nominal orbit parameters of the interceptor and the target at the initial time are given in the J2000 geocentric inertial coordinate system, and the nominal orbit parameters of the interceptor and the target at the initial time Said orbit parameters include the position vector and velocity vector of the interceptor at the initial moment and the position vector and velocity vector of the target at the initial moment; Among them, the position vector and velocity vector at the initial moment of the interceptor are recorded as r ₁₀ and v ₁₀ respectively, and the position vector and velocity vector at the initial moment of the target are respectively recorded as r ₂₀ and v ₂₀ ; Use a one-dimensional search algorithm to find the optimal energy-optimized terminal interception time within the interception time range given by the task

And calculate the pulse applied at the initial moment

The one-dimensional search algorithm is used to find the terminal interception time with optimal energy within the given interception time range of the task

And calculate the pulse applied at the initial moment

The specific process is: First, divide the interception time range [t _fmin ,t _fmax ] into N subintervals, and use the golden section method to solve the corresponding energy optimal solution in each subinterval

Finally, the energy optimal solution in the entire interception time range can be obtained by finding the minimum value among the obtained N energy optimal solutions

In the formula,

is the optimal interception time in the kth subinterval,

is the pulse applied at the corresponding initial moment in the kth subinterval,

is the terminal interception time with optimal energy within the interception time range [t _fmin ,t _fmax ],

is the pulse applied at the initial moment with optimal energy within the interception time range [t _fmin ,t _fmax ]; In the step 2, the energy optimal solution obtained in the step 1

As a reference, in the case of the navigation error at the initial moment, the error at the optimal interception moment and the error of the pulse applied at the initial moment are solved based on the optimal interception condition, and the corresponding terminal interception error is given at the same time; The specific process is: The optimal interception condition can be expressed as

in

In the formula, Δv ₁ is the velocity pulse that needs to be applied at the initial moment, Δt is the optimal interception time, v _t1 is the initial velocity vector required for the transfer orbit, and T is the transposition; At the same time, the target terminal position vector r ₂ can be expressed as r ₂ =F _t r ₂₀ +G _t v ₂₀ (3) in

In the formula, p ₂ represents the semi-radius of the target orbit,

and

Respectively represent the true anomaly angle of the target at the initial moment and the interception moment, μ represents the gravitational constant of the earth; F _t and G _t are the Lagrangian coefficients; |r ₂ | represents the size of the target terminal position vector; the pair ( 3) Calculate the partial derivative with respect to the interception time Δt, then we can get

Finally, the partial derivative of the target terminal position vector r ₂ with respect to the interception time Δt can be obtained as

in

In the formula, I ₃ represents the third-order identity matrix; True anomaly angle at the time of target interception

The partial derivative with respect to the interception time Δt is

In the formula, a ₂ and e ₂ are the semi-major axis and eccentricity of the target orbit, respectively, and E ₂ is the approach angle of the target at the moment of interception; Suppose the position and velocity navigation error of the interceptor at the initial moment is [δr ₁₀ , δv ₁₀ ], and the position and velocity navigation error of the target at the initial moment is [δr ₂₀ , δv ₂₀ ]; Then in the presence of navigation error, the speed pulse that needs to be applied at the initial moment is

In the formula, r ₁ represents the initial position vector of the interceptor, δ(Δt) represents the error of the optimal interception time, δr ₂ represents the error of the target terminal position vector;

is obtained by solving the partial derivative matrix of the Lambert problem; δr ₂ can be expressed as

In the formula, Φ ^(t) represents the two-body state transition matrix of the target from the initial moment to the interception moment; At the same time, the two-body state transition matrix can also be expressed in the form of a block matrix

In the formula,

Indicates the partial derivative of the position vector of the target interception moment to the initial moment position vector,

Represents the partial derivative of the position vector at the interception moment of the target to the velocity vector at the initial moment,

Represents the partial derivative of the velocity vector at the interception moment of the target to the position vector at the initial moment,

Indicates the partial derivative of the velocity vector at the interception moment of the target to the velocity vector at the initial moment; According to the optimal interception condition, we can get

Therefore, the error of the optimal interception time can be obtained as

in

In the formula, K represents the total derivative of the initial velocity vector v _t1 required for the transfer orbit to the optimal interception time Δt; For convenience of description, remember

In the formula, A ₁ represents the coefficient matrix of δ(Δt) affected by δr ₁₀ , A ₂ represents the coefficient matrix of δ(Δt) affected by δv ₁₀ , A ₃ represents the coefficient matrix of δ(Δt) affected by δr ₂₀ , A ₄ Represents the coefficient matrix of δ(Δt) affected by δv ₂₀ ; Then the error of the optimal interception time can be rewritten as δ(Δt)=A ₁ δr ₁₀ +A ₂ δv ₁₀ +A ₃ δr ₂₀ +A ₄ δv ₂₀ (13) The error of the pulse applied at the initial moment is

Similarly, it can be abbreviated as δ(Δv ₁ )=B ₁ δr ₁₀ +B ₂ δv ₁₀ +B ₃ δr ₂₀ +B ₄ δv ₂₀ (15) in

In the formula, B ₁ represents the coefficient matrix of δ(Δv ₁ ) affected by δr ₁₀ , B ₂ represents the coefficient matrix of δ(Δv ₁ ) affected by δv ₁₀ , B ₃ represents the coefficient matrix of δ(Δv ₁ ) affected by δr ₂₀ , B ₄ represents the coefficient matrix of δ(Δv ₁ ) affected by δv ₂₀ ; In the end, in the presence of navigation errors, the relative position error between the interceptor and the target at the time of interception is

in

In the formula

Indicates the position vector of the interceptor at the moment of interception,

Represents the position error vector of the interceptor at the moment of interception, C ₁ represents the coefficient matrix of δ(Δr _f ) affected by δr ₁₀ , C ₂ represents the coefficient matrix of δ(Δr _f ) affected by δv ₁₀ , C ₃ represents the coefficient matrix of δ(Δr _f ) is the coefficient matrix affected by δr ₂₀ , C ₄ represents the coefficient matrix of δ(Δr _f ) affected by δv ₂₀ ; Therefore, if the navigation error covariance matrix of the interceptor at a given initial moment is

The navigation error covariance matrix of the target is

Then the error variance at the optimal interception moment is:

The error standard deviation at the optimal interception moment is:

The error covariance matrix of the pulse applied at the initial moment is

The standard deviation of the applied pulse amplitude at the initial moment is

In the formula, trace() represents the trace of the solution matrix; The error covariance matrix of the relative position of the terminal is

The standard deviation of the terminal relative distance is

In the formula, E[ ] represents mathematical expectation; The terminal interception error is specifically the relative distance error between the interceptor and the target at the optimal interception moment; In said step 3, after the application time _t1 of the correction pulse is given, an analytical method is used to estimate the amplitude of the applied correction pulse and the corrected terminal intercept error, so as to determine the corrected pulse amplitude, the corrected terminal intercept error and the correction The analytical relationship at time t ₁ ; the specific process is: Assume that the position and velocity navigation errors of the interceptor and the target at the time of correction are [δr ₁₁ , δv ₁₁ ] and [δr ₂₁ , δv ₂₁ ], respectively; At this time, according to the position and velocity navigation errors of the interceptor and the target obtained at the correction time are [δr ₁₁ , δv ₁₁ ] and [δr ₂₁ , δv ₂₁ ] respectively, it can be predicted that the position error of the target at the terminal interception time is

In the formula,

is the state transition matrix of the target from the correction time t ₁ to the interception time t _f ;

is the partial derivative matrix of the position vector at the interception time of the target to the position vector at the correction time,

is the partial derivative matrix of the position vector at the interception moment of the target to the velocity vector at the correction moment; Assuming that the corrected pulse vector applied at time t ₁ is q, the position and velocity navigation errors of the interceptor and the target obtained at the corrected time are [δr ₁₁ , δv ₁₁ ] and [δr ₂₁ , δv ₂₁ ] to calculate the interceptor at The position error at the moment of terminal interception is

In the formula,

is the state transition matrix of the interceptor from the modification time t ₁ to the interception time t _f ;

is the partial derivative matrix of the position vector at the interception time of the interceptor to the position vector at the correction time,

is the partial derivative matrix of the position vector at the interception time of the interceptor to the velocity vector at the correction time; Since the purpose of applying the correction pulse is to make the interceptor and the target at the same position at the moment of interception, that is, the following conditions are met

Then the applied correction pulse can be calculated as

The above formula can be abbreviated as

in

where D ₁ represents the coefficient matrix of q affected by δr ₁₀ ; D ₂ represents the coefficient matrix of q affected by δv ₁₀ ; D ₃ represents the coefficient matrix of q affected by δr ₂₀ ; D ₄ represents the coefficient of q affected by δv ₂₀ matrix; D ₅ represents the coefficient matrix of q affected by δr ₁₁ ; D ₆ represents the coefficient matrix of q affected by δv ₁₁ ; D ₇ represents the coefficient matrix of q affected by δr ₂₁ ; D ₈ represents the coefficient matrix of q affected by δv ₂₁ ; Then the covariance matrix of the applied correction pulse is P _q ＝E[qq ^T ] The standard deviation of the applied correction pulse amplitude is

At the optimal interception moment, the real relative position error of the interceptor and the target can be expressed as

In the formula,

is the position error of the interceptor at the moment of interception after the "inaccurate" initial pulse and correction pulse are applied to the real initial state of the interceptor; Then the covariance matrix of the terminal relative position error after applying the correction is

The standard deviation of the terminal relative distance after applying the correction is

In summary, the correction pulse q applied at time t ₁ and the terminal relative position error after correction are obtained

And calculate the standard deviation of the corresponding corrected pulse amplitude σ(q) and the corrected standard deviation of the terminal interception error

In the step 4, the moment of applying the correction pulse or the corresponding time range is determined; the specific process is: According to the results in step 3, it can be seen that

and σ(q) are both unary functions of the correction pulse application time t ₁ , which can be expressed as

In the formula, h ₁ () represents the functional relationship between the terminal interception error standard deviation and the correction pulse application time, and h ₂ () represents the functional relationship between the correction pulse amplitude standard deviation and the correction pulse application time; Therefore, the weighted index J can be designed as

Where J represents the weighted index, w ₁ and w ₂ represent the weight coefficients of the two indexes respectively, and satisfy w ₁ +w ₂ =1,

and

is the normalization parameter; Finally, the weighting index can be expressed as a unary function of the moment of applying the correction pulse, that is, J=h ₃ (t ₁ ), and then a one-dimensional search algorithm can be used to determine the moment of applying the correction pulse that makes the weighting index J optimal; In the formula, h ₃ () represents the functional relationship between the weighting index J and the correction pulse application time t ₁ ; In the step 4, the moment of applying the correction pulse or the corresponding time range is determined; the specific process is: The relative accuracy requirement for a given task is

Accuracy requirements, set

has an upper precision limit of

because

is a unary function of the pulse correction time; thus, the equation can be solved by dichotomy

The zero root of , so as to obtain the corrected pulse application time range that meets the requirements of terminal interception accuracy; In the formula, h ₁ () represents the functional relationship between the terminal interception error standard deviation and the correction pulse application time.

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